\righthead{Guerra and Biondi}
\lefthead{Prestack exploding reflector modeling}
\footer{SEP--134}
\title{Prestack exploding reflector modeling: the crosstalk problem}
\author{Claudio Guerra and Biondo Biondi}

\begin{abstract}
The recently introduced prestack exploding reflector modeling aims to synthesize a small dataset comprised by areal shots, while keeping the correct kinematics to be used in iterations of migration velocity analysis. To achieve this goal, the modeled areal data must be combined into sets. This procedure generates data which is subjected to crosstalk during migration. Here, using a simple constant velocity model, we describe the crosstalk and present alternatives to mitigate it.
\end{abstract}
 
%Fortunately, by using the linearity which governs the wave propagation, the shots can be combined together into one single experiment as a way to decrease the cost of migration. This linear combination of shots, however, has the drawback of crosstalk of unrelated shots during imaging. To mitigate this problem the combination includes some type of encoding of the shots. Plane wave and phase encoding are the most popular methods. Recently, it was introduced a prestack generalization of the exploding reflector model which concept can be used to drastically reduce the amount of data to be migrated. Similarly to other methods, the migration of combined data using the concept of prestack exploding reflector model also presents crosstalk. Here, we first approach this problem by using the concept of phase encoding during the modeling step. 

\section{Introduction} 
In very complex areas, wave-equation migration is mandatory to obtain reasonable images. However, because of its high cost and small flexibility compared to Kirchhoff migration, it is rarely used in migration velocity analysis and ray-based methods dominate. In areas where the high-frequency approximation is violated and the wavefields become too complex, the use of ray-based methods might result in inaccurate velocity models that yield poor quality or unreliable images even when a state-of-the-art wave-equation algorithm is used to produce the final image. In these areas it is therefore desirable to use wave-equation migration to generate image gathers in order to evaluate the accuracy of the velocity model.
\par
Every iteration in velocity model definition for depth migration includes: 1) migration with the current velocity model; 2) measurement of some property that diagnoses the accuracy of the velocity model used in step 1; and, 3) update of the velocity model. Conventionally, Kirchhoff migration is used in step 1 and steps 2 and 3 are accomplished by measuring the departures of reflectors from the horizontal in the common image gathers and by back-projecting them along rays onto the model space, respectively. In areas where the ray theory fails, the wavefields become too complex, giving rise to multi-pathing and shadow zones, and there will be problems in updating the velocity model using ray methods.
\par
Wave-equation methods for both migration and velocity update overcome the limitations of ray-based methods \cite[]{sava04,sava03}, but at the expenses of a high cost. The practical implementation of wave-equation methods, therefore, still depends on an efficient migration/modeling scheme.
\par
Based on the linearity of wavefield extrapolators, different strategies have been formulated to reduce the computational cost of wave-equation migration. The synthesis of plane waves was first introduced by \cite{Schultz}. \cite{Rietveld01} used pre-defined wavefields of arbitrary shape to synthesize areal shots after propagation back to the surface. \cite{Rietveld02} proposed a depth migration method using the concept of controlled illumination, which can be considered as a generalization of Schultz and Claerbout's method. Phase encoded shot records \cite[]{romero} is a technique that provides the ability to combine shot records with minimum  cross-talk. The combination of phase encoding and plane-wave areal shots can be used to drastically decrease the computational effort \cite[]{Sun01}. \cite{liu} provide a general framework to evaluate plane-wave composition in prestack source plane-wave migration and show the equivalence of shot profile and plane-wave migrations.
\par
Recently, \cite{Biondi.sep.124.biondo1,Biondi.sep.131.biondo1} introduced the concept of the prestack exploding reflector modeling. This method synthesizes source and receiver wavefields at the surface, starting from a prestack migrated image cube represented by subsurface-offset domain common- gathers (SODCIGs). For the case of migration velocity analysis, it aims to generate a smaller dataset than the one used in the initial migration while keeping the kinematics necessary to perform migration velocity analysis.
\par
If the migration velocity is correct and assuming sufficiently good illumination, the reflectors are focused around the zero subsurface offset and the method reduces to the conventional exploding reflector modeling. However, when the migration velocity is inaccurate and in the presence of dip, a pre-processing step is needed before modeling. \cite{Biondi.sep.131.biondo1} describes how to generate dip-independent image gathers to be used as initial condition for the upward propagation of source and receiver wavefields. This is achieved by rotating the SODCIGs according to the geological dip in the wavenumber domain.
\par
 Conceptually, one SODCIG is used as an initial condition to upward propagate source and receiver wavefields to be recorded at the surface by an areal shot along the entire survey. Depending on the relationship between shot spacing and SODCIG spacing, this procedure can actually generate more individual experiments than the existing in the original data. However, we take advantage of the linearity of the wave propagation to combine several experiments into a set of experiments, therefore decreasing the amount of data to migrate.
\par
The combination of several experiments gives rise to crosstalk during imaging. \cite{Biondi.sep.124.biondo1} uses a decorrelation distance between SODCIGs sufficiently large to prevent crosstalk. By using this criterion, however, the maximum data reduction is  limited to the ratio between the number of SODCIGs and the decorrelation distance. Furthermore, he mentions that an additional saving can be achieved by using concepts similar to phase encoding \cite[]{romero} during the modeling step, allowing to combine more shots into one set. From now on, upward propagated areal shots combined into one single areal shot will be called a set.
\par
In the aforementioned papers, \cite{Biondi.sep.124.biondo1,Biondi.sep.131.biondo1} uses the two-way wave equation to propagate the wavefields. Here, we use the one-way wave equation to achieve additional computational saving. Furthermore, under the one-way framework, we are enabled to introduce a phase encoding like scheme during the modeling to reduce cross-talk during migration while applying the imaging condition.

\section{Prestack exploding reflector modeling}
In 2D, the modeling of source and receiver wavefields at $z=0$, using the one-way wave equation and starting from a prestack image at a selected position, $x_{\xi}$, can be described by:
\begin{equation}
\begin{array}{cc}
S(x,\omega) = G(z,x_{\xi}-h_{\xi};x,z=0,\omega)*I_s(z,x_{\xi},h_{\xi}), \\
R(x,\omega) = G(z,x_{\xi}+h_{\xi};x,z=0,\omega)*I_r(z,x_{\xi},h_{\xi}), 
\end{array}
\label{eq:eq1}
\end{equation}
where $S(x,\omega)$ is the source wavefield at $z=0$; $R(x,\omega)$ is the receiver wavefield at $z=0$; $I_s(z,x_{\xi},h_{\xi})$ and $I_r(z,x_{\xi},h_{\xi})$ are the prestack images used as initial condition for the source and receiver wavefield extrapolation, respectively; $G(z,x_{\xi};x,z=0,\omega)$ represents the one-way operator which extrapolates the wavefields from the subsurface to the surface; $h_{\xi}$ is the subsurface offset; $z_{\xi}$ is depth; $\omega$ is the temporal frequency and $x$ is the spatial coordinate in the data space which, in this case, coincides with $x_{\xi}$. Notice that for a perfectly focused image, $h_{\xi}=0$ and equation \ref{eq:eq1} reduces to the conventional zero-offset exploding reflector modeling. The prestack images used as initial condition for the source and receiver wavefield extrapolation are supposed to be dip-independent gathers computed by changing the dip along the offset direction according to the apparent geological dip \cite[]{Biondi.sep.131.biondo1}.
\par
The number of individual experiments described in equation \ref{eq:eq1} equals the number of shots used in the initial migration if the CMP spacing is equal to the shot spacing. Therefore, to decrease the amount of input data to migration, individual experiments can be combined together into sets of areal shots after the upward propagation. This is achieved by regularly selecting individual experiments and adding them up into their specific set, after being upward propagated, according to:
\begin{eqnarray}
\tilde{S}(x,\omega,n)& = &\sum_{i=1}^n C_i(x)S_(x,\omega), \nonumber \\
\tilde{R}(x,\omega,n)& = &\sum_{i=1}^n C_i(x)R_(x,\omega), 
\label{eq:eq2}
\end{eqnarray}
where $\tilde{S}(x,\omega,n)$ represents the $n$ sets of combined experiments for the source wavefield; $\tilde{R}(x,\omega,n)$ represents the $n$ sets of combined experiments for the receiver wavefield; and, $C(x)$ is the sampling function different from zero at every $n^{th}$ trace, shifted up to all experiments are sampled. Pairs of $\tilde{S}(x,\omega)$ and  $\tilde{R}(x,\omega)$ are to be used as the areal source function and the areal receiver wavefield, respectively, in areal shot migration.
\par
As we will show in the next section, during imaging by crosscorrelation, unrelated events pertaining to the same pair of sets of areal source and areal receiver correlate together, generating migration artifacts in the form of crosstalk. The crosstalk degrades the kinematics we are interested in to perform migration velocity analysis.

\section{Crosstalk generation and attenuation}

Wavefield propagation is a linear process. This allows us to linearly combine wavefields  before the propagation effects are removed by migration. Several migration methods explore this linearity to decrease the amount of data to migration \cite[]{Rietveld02, romero, liu}. However, while applying the imaging condition, energy from unrelated source and receiver wavefields crosscorrelate generating crosstalk. For the present case, let us consider the migration of a set of added areal shots containing just a pair of individual experiments. The prestack image is formed by crosscorrelation of source and receiver wavefields according to:
\begin{equation}
\tilde{I}(z_{\xi},x_{\xi},h_{\xi}) = \sum_{\omega} \tilde{S}^{\star}(z_{\xi},x_{\xi}-h_{\xi},\omega) 
\tilde{R}(z_{\xi},x_{\xi}+h_{\xi},\omega),
\label{eq:eq3}
\end{equation}
where $^{\star}$ represents complex conjugation. If $\tilde{S}(x,\omega,n)$ and $\tilde{R}(x,\omega,n)$ in equation~\ref{eq:eq2} are comprised by two summed areal shots, the image $\tilde{I}(z_{\xi},x_{\xi},h_{\xi}) $ will be given by:
\begin{eqnarray}
\lefteqn{\tilde{I}(z_{\xi},x_{\xi},h_{\xi}) = I_1(z_{\xi},x_{\xi},h_{\xi}) + I_2(z_{\xi},x_{\xi},h_{\xi}) + } \nonumber \\
 &  &  \sum_{\omega}S_1^{\star}(z_{\xi},x_{\xi}-h_{\xi},\omega)R_2(z_{\xi},x_{\xi}+h_{\xi},\omega) + \nonumber \\
 &  &  \sum_{\omega}S_2^{\star}(z_{\xi},x_{\xi}-h_{\xi},\omega)R_1(z_{\xi},x_{\xi}+h_{\xi},\omega).
\label{eq:eq4}
\end{eqnarray}
In equation \ref{eq:eq4}, the last two terms in the summations are the crosstalk terms. \cite{romero} address the problem of attenuating these migration artifacts by multiplying the source functions and the corresponding receiver wavefields by a function of frequency in such a way that the crosstalks are dispersed (random phase encoding) throughout the image after stacking over frequency, during imaging, or shifted outside of the image space (linear phase encoding). Here, we use a similar approach while modeling the areal shots.
\par
To exemplify the problem of crosstalk in the prestack exploding reflector strategy let us make use of a simple constant velocity model of 2 km/s with two intersecting reflectors, one horizontal and the other dipping 15$^{\circ}$. The dataset is comprised of 200 split-spread shots spaced every 20 m with a maximum offset of 2000 m. Figure~\ref{fig:mig08} shows the result of shot profile migration using the correct velocity. The trace spacing in the migrated result is 20 m and the number of subsurface offsets in the prestack image is 41 at every 20 m. The front panel corresponds to the zero-subsurface offset section, the side panel is a SODCIG selected at x=1.5 km and the upper panel is a constant depth slice. This data will be used to model the areal data.
\sideplot{mig08}{width=3.5in}{Prestack cube obtained with shot profile migration with the correct velocity.}
\par
As migration was performed with the correct velocity, the pre-processing step of rotating the SODCIGs according to the geological dip is not necessary before modeling. Therefore, to model the areal sources and the areal receivers, SODCIGs are upward propagated, using the one-way wave equation (equation~\ref{eq:eq1}) without any pre-processing step. The result of one modeling experiment is collected at the surface as an areal shot and assigned to its specific set. To save computational time, an aperture region around the SODCIG can be defined. 
\par
Figures \ref{fig:arsource} and \ref{fig:arrecev} show sets of areal sources and areal receivers, in time domain, computed with different distance between SODCIGs used as initial condition. Since the areal source is backward propagated it exists for negative times. For both figures, for the set of areal data labeled a) the distance between SODCIGs is 51 traces (1000 m), for the set labeled b) is 41 traces (800 m)  and for the set labeled c) is 11 (200 m). These distances in traces defines the number of sets to be migrated and, consequently, the efficiency of migration. As we show next, this  also influences the intensity of the crosstalk.
\plot{arsource}{width=6in}{Sets of areal sources computed with different distances between SODCIGs used as initial condition. a) 51 traces; b) 41 traces; and , c) 11 traces.}
\plot{arrecev}{width=6in}{Sets of areal receivers computed with different distances between SODCIGs used as initial condition. a) 51 traces; b) 41 traces; and , c) 11 traces.}
\par
The crosstalk intensity on a subsequent migration depends on the distance between SODCIGs selected to make up a set of areal data. \cite{Biondi.sep.124.biondo1} shows that, using a decorrelation distance greater than twice the maximum subsurface offset, the crosstalks can be eliminated for the case of non-intersecting reflectors. For the case of reflectors with different geological dip and, moreover, intersecting, this procedure does not guarantee the absence of crosstalk. 
\par
Figures \ref{fig:armig51W}, \ref{fig:armig41W} and \ref{fig:armig11W} show the areal shot migration using crosscorrelation  imaging condition of the datasets which contain the sets shown in Figures \ref{fig:arsource} and \ref{fig:arrecev}. The number of subsurface offsets is 41. Figure \ref{fig:armig51W} is the migration of 51 sets (Figure \ref{fig:arsource}a and \ref{fig:arrecev}a) , Figure \ref{fig:armig41W} is the migration of 41 sets (Figure \ref{fig:arsource}b and \ref{fig:arrecev}b) and Figure \ref{fig:armig11W} is the migration of 11 sets (Figure \ref{fig:arsource}c and \ref{fig:arrecev}c). The migration of the data with 51 sets shows no crosstalk in the SODCIG, because the distance of 51 traces is bigger than the decorrelation distance. The migration of the data with 41 sets shows an acceptable crosstalk in the SODCIG and the crosstalk is strong in the SODCIG of the data with 11 sets. Notice that for all of the three results a strong crosstalk occurs at zero-subsurface offset (x,z-domain). 
\sideplot{armig51W}{width=3.5in}{Areal shot migration of 51 sets, each set containing the summation of areal shots initiated at every $51^{th}$ SODCIG.}
\sideplot{armig41W}{width=3.5in}{Areal shot migration of 41 sets, each set containing the summation of areal shots initiated at every $41^{th}$ SODCIG.}
\sideplot{armig11W}{width=3.5in}{Areal shot migration of 11 sets, each set containing the summation of areal shots initiated at every $11^{th}$ SODCIG.}
\par
The crosstalks described in the previous paragraph have two distinct origins. The one which occurs in the zero-subsurface offset section as a reflector with an intermediate dip results from the croscorrelation of reflections in the source wavefield  with reflections in the receiver wavefield from a single upward propagation experiment. The crosscorrelation occurs at times different from zero propagation time. Therefore, one possible strategy is to time-windowing the application of imaging condition around zero propagation time.  This strategy will be described in the next paragraph. The crosstalk present in offsets different from zero in the SODCIGs has similar origin as the one described by \cite{sava02}. In the case of the prestack exploding reflector modeling, they are related to reflections in the source and receiver wavefields pertaining to different areal data summed together into the same set. \cite{sava02} uses an imaging condition which crosscorrelates decomposed source and receiver wavefields as a function of local slope at every position and time. This procedure yields good results, but at the expenses of a high computational cost. 
\par
According to the exploding reflector model \cite[]{loewenthal}, reflectors explode at time zero. Therefore, we can take advantage that the areal source wavefield also contains reflections that should be focused around zero time of wavefield propagation. As \cite{Biondi.sep.131.biondo1} shows, the crosscorrelation of areal wavefields within a small window around zero time of wavefield propagation can diminish the crosstalk at zero-subsurface offset. Equation \ref{eq:eq5} shows that, using the time-windowed imaging condition, for the zero-subsurface offset, crosstalk only occurs between events which have similar traveltimes.
\begin{eqnarray}
\lefteqn{\tilde{I}(z_{\xi},x_{\xi},h_{\xi}=0) = I_1(z_{\xi},x_{\xi},h_{\xi}=0) + I_2(z_{\xi},x_{\xi},h_{\xi}=0) + }   \nonumber \\
 &  &  \sum_{-\Delta t \leq t=0 \leq \Delta t}S_1(z_{\xi},x_{\xi},t)R_2(z_{\xi},x_{\xi},t) + \nonumber \\
 &  &  \sum_{-\Delta t \leq t=0 \leq \Delta t}S_2(z_{\xi},x_{\xi},t)R_1(z_{\xi},x_{\xi},t).
\label{eq:eq5}
\end{eqnarray}
\par
To apply this imaging condition using an one-way propagator, the wavefields must be kept in memory for all the frequencies which increases the memory requirements. On the other hand, the application of the imaging condition for a small number of samples in time saves much computational effort. Figures \ref{fig:armig51T}, \ref{fig:armig41T} and \ref{fig:armig11T} show the areal shot migration using the time-windowed crosscorrelation imaging condition from equation \ref{eq:eq4}. Again, the number of subsurface offsets is 41. Figure \ref{fig:armig51T} is the migration of 51 sets, Figure \ref{fig:armig41T} is the migration of 41 sets and Figure \ref{fig:armig11T} is the migration of 11 sets. Notice how the crosstalk in the zero-subsurface offset section has been largely attenuated in all of the three migration results. 
\sideplot{armig51T}{width=3.5in}{Areal shot migration using time-windowed imaging condition of 51 sets, each set containing the summation of areal shots initiated at every $51^{th}$ SODCIG. Compare with Figure \ref{fig:armig51W}.}
\sideplot{armig41T}{width=3.5in}{Areal shot migration using time-windowed imaging condition of 41 sets, each set containing the summation of areal shots initiated at every $41^{th}$ SODCIG. Compare with Figure \ref{fig:armig41W}.}
\sideplot{armig11T}{width=3.5in}{Areal shot migration using time-windowed imaging condition of 11 sets, each set containing the summation of areal shots initiated at every $11^{th}$ SODCIG. Compare with Figure \ref{fig:armig11W}.}
\par
The time-windowed imaging condition, however, does not attenuate the crosstalk in the SODCIG and for the migration of Figure \ref{fig:armig11T} it is still severe. Ideally, the areal shots pertaining to the same set should be uncorrelated. A possible way to decrease the correlation between areal shots is by encoding them with a random phase function, $a(z,x,\omega)$, in such a way that the crosstalk is dispersed throughout the image. As \cite{romero} show, for the random phase encoding we can choose $a(z,x,\omega) = e^{i f(z,x,\omega)}$, where $f(z,x,\omega)$ is a random sequence as a function of z, x and $\omega$.
\par
The modeling with phase encoding synthesizes data according to
\begin{equation}
\begin{array}{cc}
S_a(x,\omega) = a(z,x,\omega) G(z_{\xi},x_{\xi}-h_{\xi};x,z=0,\omega)*I_s(z_{\xi},x_{\xi},h_{\xi}), \\
R_a(x,\omega) = a(z,x,\omega) G(z_{\xi},x_{\xi}+h_{\xi};x,z=0,\omega)*I_r(z_{\xi},x_{\xi},h_{\xi}), 
\end{array}
\label{eq:eq6}
\end{equation}
and the sets of random phase encoded areal sources and receivers, $\tilde{S_a}(x,\omega,n)$ and $\tilde{R_a}(x,\omega,n)$, respectively, are given by:
\begin{eqnarray}
\tilde{S_a}(x,\omega,n)& = &\sum_{i=1}^n C_i(x)S_a(x,\omega), \nonumber \\
\tilde{R_a}(x,\omega,n)& = &\sum_{i=1}^n C_i(x)R_a(x,\omega), 
\label{eq:eq7}
\end{eqnarray}
\par
Figures \ref{fig:arrphz}a and \ref{fig:arrphz}b show one pair of set of random phase encoded areal source and receiver, respectively. Alternatively, the encoding can be performed just in $x$ and $\omega$. Figures \ref{fig:arrphw}a and \ref{fig:arrphw}b show one pair of set of $(x,\omega)$-random phase encoded areal source and receiver, respectively. 

\plot{arrphz}{width=5in}{Areal source (a) and receiver (b) computed with random phase encoding, $a(z,x,\omega)$.}
\plot{arrphw}{width=5in}{Areal source (a) and receiver (b) computed with random phase encoding, $a(x,\omega)$.}

Equation \ref{eq:eq8} refers to an image after areal shot migration of one set comprised by two random phase encoded areal sources and receivers:
\begin{eqnarray}
\lefteqn{\tilde{I}(z_{\xi},x_{\xi},h_{\xi}) = I_1(z_{\xi},x_{\xi},h_{\xi}) + I_2(z_{\xi},x_{\xi},h_{\xi}) + } \nonumber \\
 &  &  \sum_{\omega}a_1^{\star}a_2 S_{a_1}^{\star}(z_{\xi},x_{\xi}-h_{\xi},\omega)R_{a_2}(z_{\xi},x_{\xi}+h_{\xi},\omega) + \nonumber \\
 &  &  \sum_{\omega}a_2^{\star}a_1 S_{a_2}^{\star}(z_{\xi},x_{\xi}-h_{\xi},\omega)R_{a_1}(z_{\xi},x_{\xi}+h_{\xi},\omega).
\label{eq:eq8}
\end{eqnarray}
The crosscorrelation between the phase encoding functions $a_1(z,x,\omega)$ and $a_2(z,x,\omega)$ must provide random values which will disperse the crosstalk by the image.
\par
Figures \ref{fig:arm11phw} and \ref{fig:arm11phz} show the areal shot migration results of 11 sets of randomly encoded areal data using encoding functions which vary in $(x,\omega)$ and in $(z,x,\omega)$, respectively. Comparison between Figure \ref{fig:arm11phw} and Figure \ref{fig:armig11W} shows that just encoding in $(x,\omega)$ yields a significant reduction of the crosstalk in the SODCIG. The crosstalk in the zero-subsurface offset, however, is much less affected. Comparison between Figure \ref{fig:arm11phz} and Figure \ref{fig:armig11W} shows that  encoding in $(z,x,\omega)$ performs as efficiently as encoding in $(x,\omega)$ and, moreover, achieves the goal of dispersing the crosstalk in the zero-subsurface offset section. 
\par
Since the number of phase encoded areal data is small, the noise amplitudes dispersed through the image are not negligible. By migrating more sets of  $(z,x,\omega)$-random phase encoded areal data, these amplitudes can be largely attenuated. Figure \ref{fig:arm11phz4} shows the superior result when compared to Figures \ref{fig:armig11W}, \ref{fig:arm11phw} and \ref{fig:arm11phz}, obtained by migrating 4 groups of 11 sets of $(z,x,\omega)$--random encoded areal data. Besides the crosstalk has been virtually eliminated, the speckled noise has lower amplitudes than in Figure \ref{fig:arm11phz}.

\sideplot{arm11phw}{width=3.5in}{Areal shot migration of random encoded data. 11 sets of $(x,\omega)$--random encoded areal data were migrated. Compare with Figure \ref{fig:armig11W}.}
\sideplot{arm11phz}{width=3.5in}{Areal shot migration of random encoded data. 11 sets of $(z,x,\omega)$--random encoded areal data were migrated. Compare with Figure \ref{fig:armig11W}.}
\sideplot{arm11phz4}{width=3.5in}{Areal shot migration of random encoded data. 4 groups of 11 sets of $(z,x,\omega)$--random encoded areal data were migrated. Compare with Figures \ref{fig:armig11W}, \ref{fig:arm11phw} and \ref{fig:arm11phz}.}
\par
One drawback of applying random phase encoding is that it makes the reflectors randomly ``explode'' at times different from the zero time of the wavefields. Consequently, the time-windowed imaging condition can not be applied. Perhaps, the linear phase encoding enable us to use the time-windowed imaging condition. This deserves future research.  

\section{Conclusions}
The prestack exploding reflector method has the potential of decrease the amount of data to perform migration velocity analysis. To be cost efficient, the method relies on the linearity of the wave-propagation to combine modeled areal shots into some supersets of areal shots. This process gives rise to crosstalk when the areal data is submitted to migration.
\par
We showed that applying a time-windowed imaging condition is efficient for the crosstalk in the zero-subsurface offset section, but does not help much in attenuating crosstalk in SODCIGs. We presented a strategy to perform random phase encoding during the modeling. The results are promising and much better than the time-windowed imaging condition regarding the attenuation of crosstalk in the SODCIGs. The next step will be the application of this strategy to data from complex geology.
\par
Because of the random encoding, the reflectors do not focus anymore close to the zero time of the wave propagation, disabling the use of the time-windowed imaging condition. Further research will investigate the use of linear phase encoding associated to the application of this imaging condition.
\par


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